We are tasked with solving the boundary value problem (BVP) where:

1. \( U_x = 4 U_y \) (a first-order partial differential equation).
2. \( U(0, y) = 8e^{-3y} \) (the boundary condition).

### Step 1: Analyze the PDE
The given partial differential equation is:
\[
U_x = 4U_y
\]
This can be rewritten as:
\[
\frac{\partial U}{\partial x} = 4 \frac{\partial U}{\partial y}
\]
This suggests that the function \( U(x, y) \) depends on a combination of both \( x \) and \( y \). To solve this equation, we employ the method of characteristics.

### Step 2: Characteristic equations
The characteristic equations for this PDE are derived from the ratios of the differentials:
\[
\frac{dx}{1} = \frac{dy}{4} = \frac{dU}{0}
\]
This gives us two key relationships:
- \( \frac{dx}{dy} = 1/4 \), which implies that \( x - \frac{y}{4} = C \), where \( C \) is a constant.
- The equation \( \frac{dU}{0} = 0 \) implies that \( U \) is constant along the characteristic curves.

Thus, the general solution for \( U(x, y) \) will be a function of the characteristic combination \( x - \frac{y}{4} \):
\[
U(x, y) = f\left( x - \frac{y}{4} \right)
\]
where \( f \) is some arbitrary function to be determined.

### Step 3: Apply the boundary condition
We are given the boundary condition:
\[
U(0, y) = 8e^{-3y}
\]
Substituting \( x = 0 \) into the general solution:
\[
U(0, y) = f\left( 0 - \frac{y}{4} \right) = f\left( -\frac{y}{4} \right)
\]
This must equal \( 8e^{-3y} \), so we have:
\[
f\left( -\frac{y}{4} \right) = 8e^{-3y}
\]
To find \( f(z) \), let \( z = -\frac{y}{4} \). Then \( y = -4z \), and the equation becomes:
\[
f(z) = 8e^{12z}
\]

### Step 4: Final solution
Therefore, the solution to the PDE is:
\[
U(x, y) = f\left( x - \frac{y}{4} \right) = 8e^{12\left( x - \frac{y}{4} \right)}
\]
Simplifying:
\[
U(x, y) = 8e^{12x - 3y}
\]

### Final Answer:
The solution to the boundary value problem is:
\[
U(x, y) = 8e^{12x - 3y}
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:
1. How does the method of characteristics apply to linear first-order PDEs?
2. What is the physical meaning of the boundary condition \( U(0, y) = 8e^{-3y} \)?
3. Can you apply the method of separation of variables to this type of PDE?
4. How does the solution \( U(x, y) = 8e^{12x - 3y} \) behave as \( x \) and \( y \) increase?
5. What other boundary conditions could make this problem more complex?

**Tip:** Always verify the solution by substituting it back into the original PDE and boundary conditions to check consistency.

Explore the detailed solution of the boundary value problem Ux = 4Uy with the boundary condition u(0, y) = 8e^{-3y}. This problem is solved using the method of characteristics and involves finding the general solution U(x, y) = 8e^{12x - 3y}. Suitable for undergraduate mathematics students studying partial differential equations.

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